Does a faerie dragon familiar's spellcasting advance when the PC levels up?

Question

If a PC has the Improved Familiar feat and chooses a faerie dragon, does the faerie dragon's spellcasting (it is listed as casting spells as a 3rd level Sorcerer) advance when the PC levels up?

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A player in my Pathfinder group has this build and claims that the faerie dragon familiar is treated as a sorcerer of equal level to its master; this seems terribly unbalanced (e.g., having a matched-level full arcane caster for the cost of 1 feat) compared to, say, Leadership (as a cohort is capped at 2 levels below your own), but I have not found any explicit text or rule(s) to settle this.

Answer 1

A faerie dragon's spellcasting does not advance with the PC

Improved Familiar doesn't change any basics about how familiars work.¹ Your player is probably thinking about (and misinterpreting) the part of the familiar rules about abilities related to HD:

Hit Dice: For the purpose of effects related to number of Hit Dice, use the master's character level or the familiar's normal HD total, whichever is higher.

Importantly for the faerie dragon, its spellcasting is not an ability related to its hit dice. It has a fixed ability to cast spells as a 3rd-level sorcerer, which — as a normal monster — is unrelated to its hit dice. Becoming a familiar doesn't change this, and its spellcasting ability remains unrelated to its hit dice. As a result, its spellcasting isn't “effects related to number of Hit Dice” and can't benefit from the rule that familiars can calculated HD-related abilities using their master's level.

^{1. With two exceptions — about the familiar's creature type and speech abilities — which aren't relevant here.}